Nonlinear Analysis:Real World Applications7(2006)1104– 1118 https://www.doczj.com/doc/7b18086411.html,/locate/na Analysis of a predator–prey model with modi?ed Leslie–Gower and Holling-type II schemes with time delay A.F.Nindjin a,M.A.Aziz-Alaoui b,?,M.Cadivel b a Laboratoire de Mathématiques Appliquées,Universitéde Cocody,22BP582,Abidjan22,C?te d’Ivoire,France b Laboratoire de Mathématiques Appliquées,Universitédu Havre,25rue Philippe Lebon,B.P.540,76058Le Havre Cedex,France Received17July2005;accepted7October2005 Abstract Two-dimensional delayed continuous time dynamical system modeling a predator–prey food chain,and based on a modi?ed version of Holling type-II scheme is investigated.By constructing a Liapunov function,we obtain a suf?cient condition for global stability of the positive equilibrium.We also present some related qualitative results for this system. ?2005Elsevier Ltd.All rights reserved. Keywords:Time delay;Boundedness;Permanent;Local stability;Global stability;Liapunov functional 1.Introduction The dynamic relationship between predators and their prey has long been and will continue to be one of dominant themes in both ecology and mathematical ecology due to its universal existence and importance.A major trend in theoretical work on prey–predator dynamics has been to derive more realistic models,trying to keep to maximum the unavoidable increase in complexity of their mathematics.In this optic,recently[2],see also[1,5,6]has proposed a?rst study of two-dimensional system of autonomous differential equation modeling a predator prey system.This model incorporates a modi?ed version of Leslie–Gower functional response as well as that of the Holling-type II. They consider the following model ???? ???˙x= a1?bx? c1y x+k1 x, ˙y= a2? c2y x+k2 y (1) with the initial conditions x(0)>0and y(0)>0. This two species food chain model describes a prey population x which serves as food for a predator y. The model parameters a1,a2,b,c1,c2,k1and k2are assuming only positive values.These parameters are de?ned as follows:a1is the growth rate of prey x,b measures the strength of competition among individuals of species x,c1 ?Corresponding author.Tel./fax:+133232744. E-mail address:Aziz-Alaoui@univ-lehavre.fr(M.A.Aziz-Alaoui). 1468-1218/$-see front matter?2005Elsevier Ltd.All rights reserved. doi:10.1016/j.nonrwa.2005.10.003
脐带间充质干细胞的研究进展 间充质干细胞（mesenchymal stem cells,MSC S ）是来源于发育早期中胚层 的一类多能干细胞[1-5]，MSC S 由于它的自我更新和多项分化潜能，而具有巨大的 治疗价值 ,日益受到关注。MSC S 有以下特点：(1)多向分化潜能，在适当的诱导条件下可分化为肌细胞[2]、成骨细胞[3、4]、脂肪细胞、神经细胞[9]、肝细胞[6]、心肌细胞[10]和表皮细胞[11, 12]；(2)通过分泌可溶性因子和转分化促进创面愈合；(3) 免疫调控功能，骨髓源（bone marrow ）MSC S 表达MHC-I类分子，不表达MHC-II 类分子，不表达CD80、CD86、CD40等协同刺激分子，体外抑制混合淋巴细胞反应，体内诱导免疫耐受[11, 15]，在预防和治疗移植物抗宿主病、诱导器官移植免疫耐受等领域有较好的应用前景；(4)连续传代培养和冷冻保存后仍具有多向分化潜能，可作为理想的种子细胞用于组织工程和细胞替代治疗。1974年Friedenstein [16] 首先证明了骨髓中存在MSC S ，以后的研究证明MSC S 不仅存在于骨髓中，也存在 于其他一些组织与器官的间质中：如外周血[17]，脐血[5]，松质骨[1, 18]，脂肪组织[1]，滑膜[18]和脐带。在所有这些来源中，脐血(umbilical cord blood)和脐带(umbilical cord)是MSC S 最理想的来源，因为它们可以通过非侵入性手段容易获 得，并且病毒污染的风险低，还可冷冻保存后行自体移植。然而，脐血MSC的培养成功率不高[19, 23-24]，Shetty 的研究认为只有6%，而脐带MSC的培养成功率可 达100%[25]。另外从脐血中分离MSC S ，就浪费了其中的造血干/祖细胞(hematopoietic stem cells/hematopoietic progenitor cells,HSCs/HPCs) [26, 27]，因此，脐带MSC S (umbilical cord mesenchymal stem cells, UC-MSC S )就成 为重要来源。 一．概述 人脐带约40 g, 它的长度约60–65 cm, 足月脐带的平均直径约1.5 cm[28, 29]。脐带被覆着鳞状上皮，叫脐带上皮，是单层或复层结构，这层上皮由羊膜延续过来[30, 31]。脐带的内部是两根动脉和一根静脉，血管之间是粘液样的结缔组织，叫做沃顿胶质，充当血管外膜的功能。脐带中无毛细血管和淋巴系统。沃顿胶质的网状系统是糖蛋白微纤维和胶原纤维。沃顿胶质中最多的葡萄糖胺聚糖是透明质酸，它是包绕在成纤维样细胞和胶原纤维周围的并维持脐带形状的水合凝胶，使脐带免受挤压。沃顿胶质的基质细胞是成纤维样细胞[32]，这种中间丝蛋白表达于间充质来源的细胞如成纤维细胞的，而不表达于平滑肌细胞。共表达波形蛋白和索蛋白提示这些细胞本质上肌纤维母细胞。 脐带基质细胞也是一种具有多能干细胞特点的细胞，具有多项分化潜能，其 形态和生物学特点与骨髓源性MSC S 相似[5, 20, 21, 38, 46]，但脐带MSC S 更原始，是介 于成体干细胞和胚胎干细胞之间的一种干细胞，表达Oct-4, Sox-2和Nanog等多
精神分裂症的病因及发病机理 精神分裂症病因：尚未明，近百年来的研究结果也仅发现一些可能的致病因素。（一）生物学因素1．遗传遗传因素是精神分裂症最可能的一种素质因素。国内家系调查资料表明：精神分裂症患者亲属中的患病率比一般居民高6.2倍，血缘关系愈近，患病率也愈高。双生子研究表明：遗传信息几乎相同的单卵双生子的同病率远较遗传信息不完全相同 的双卵双生子为高，综合近年来11项研究资料：单卵双生子同病率（56.7%），是双卵双生子同病率（12.7%）的4.5倍，是一般人口患难与共病率的35-60倍。说明遗传因素在本病发生中具有重要作用，寄养子研究也证明遗传因素是本症发病的主要因素，而环境因素的重要性较小。以往的研究证明疾病并不按类型进行遗传，目前认为多基因遗传方式的可能性最大，也有人认为是常染色体单基因遗传或多源性遗传。Shields发现病情愈轻，病因愈复杂，愈属多源性遗传。高发家系的前瞻性研究与分子遗传的研究相结合，可能阐明一些问题。国内有报道用人类原癌基因Ha-ras-1为探针，对精神病患者基因组进行限止性片段长度多态性的分析，结果提示11号染色体上可能存在着精神分裂症与双相情感性精神病有关的DNA序列。2．性格特征：约40%患者的病前性格具有孤僻、冷淡、敏感、多疑、富于幻想等特征，即内向
型性格。3．其它：精神分裂症发病与年龄有一定关系，多发生于青壮年，约1/2患者于20～30岁发病。发病年龄与临床类型有关，偏执型发病较晚，有资料提示偏执型平均发病年龄为35岁，其它型为23岁。80年代国内12地区调查资料：女性总患病率（7.07%。）与时点患病率（5.91%。）明显高于男性（4.33%。与3.68%。）。Kretschmer在描述性格与精神分裂症关系时指出：61%患者为瘦长型和运动家型，12.8%为肥胖型，11.3%发育不良型。在躯体疾病或分娩之后发生精神分裂症是很常见的现象，可能是心理性生理性应激的非特异性影响。部分患者在脑外伤后或感染性疾病后发病；有报告在精神分裂症患者的脑脊液中发现病毒性物质；月经期内病情加重等躯体因素都可能是诱发因素，但在精神分裂症发病机理中的价值有待进一步证实。（二）心理社会因素1．环境因素①家庭中父母的性格，言行、举止和教育方式（如放纵、溺爱、过严）等都会影响子女的心身健康或导致个性偏离常态。②家庭成员间的关系及其精神交流的紊乱。③生活不安定、居住拥挤、职业不固定、人际关系不良、噪音干扰、环境污染等均对发病有一定作用。农村精神分裂症发病率明显低于城市。2．心理因素一般认为生活事件可发诱发精神分裂症。诸如失学、失恋、学习紧张、家庭纠纷、夫妻不和、意处事故等均对发病有一定影响，但这些事件的性质均无特殊性。因此，心理因素也仅属诱发因
脐带血造血干细胞库管理办法（试行） 第一章总则 第一条为合理利用我国脐带血造血干细胞资源，促进脐带血造血干细胞移植高新技术的发展，确保脐带血 造血干细胞应用的安全性和有效性，特制定本管理办法。 第二条脐带血造血干细胞库是指以人体造血干细胞移植为目的，具有采集、处理、保存和提供造血干细胞 的能力，并具有相当研究实力的特殊血站。 任何单位和个人不得以营利为目的进行脐带血采供活动。 第三条本办法所指脐带血为与孕妇和新生儿血容量和血循环无关的，由新生儿脐带扎断后的远端所采集的 胎盘血。 第四条对脐带血造血干细胞库实行全国统一规划，统一布局，统一标准，统一规范和统一管理制度。 第二章设置审批 第五条国务院卫生行政部门根据我国人口分布、卫生资源、临床造血干细胞移植需要等实际情况，制订我 国脐带血造血干细胞库设置的总体布局和发展规划。 第六条脐带血造血干细胞库的设置必须经国务院卫生行政部门批准。 第七条国务院卫生行政部门成立由有关方面专家组成的脐带血造血干细胞库专家委员会（以下简称专家委
员会），负责对脐带血造血干细胞库设置的申请、验收和考评提出论证意见。专家委员会负责制订脐带血 造血干细胞库建设、操作、运行等技术标准。 第八条脐带血造血干细胞库设置的申请者除符合国家规划和布局要求，具备设置一般血站基本条件之外， 还需具备下列条件： （一）具有基本的血液学研究基础和造血干细胞研究能力； （二）具有符合储存不低于1 万份脐带血的高清洁度的空间和冷冻设备的设计规划； （三）具有血细胞生物学、HLA 配型、相关病原体检测、遗传学和冷冻生物学、专供脐带血处理等符合GMP、 GLP 标准的实验室、资料保存室； （四）具有流式细胞仪、程控冷冻仪、PCR 仪和细胞冷冻及相关检测及计算机网络管理等仪器设备； （五）具有独立开展实验血液学、免疫学、造血细胞培养、检测、HLA 配型、病原体检测、冷冻生物学、 管理、质量控制和监测、仪器操作、资料保管和共享等方面的技术、管理和服务人员； （六）具有安全可靠的脐带血来源保证； （七）具备多渠道筹集建设资金运转经费的能力。 第九条设置脐带血造血干细胞库应向所在地省级卫生行政部门提交设置可行性研究报告，内容包括：
精神分裂症的发病原因是什么 精神分裂症是一种精神病，对于我们的影响是很大的，如果不幸患上就要及时做好治疗，不然后果会很严重，无法进行正常的工作和生活，是一件很尴尬的事情。因此为了避免患上这样的疾病，我们就要做好预防，今天我们就请广州协佳的专家张可斌来介绍一下精神分裂症的发病原因。 精神分裂症是严重影响人们身体健康的一种疾病，这种疾病会让我们整体看起来不正常，会出现胡言乱语的情况，甚至还会出现幻想幻听，可见精神分裂症这种病的危害程度。 (1)精神刺激：人的心理与社会因素密切相关，个人与社会环境不相适应，就产生了精神刺激，精神刺激导致大脑功能紊乱，出现精神障碍。不管是令人愉快的良性刺激，还是使人痛苦的恶性刺激，超过一定的限度都会对人的心理造成影响。 (2)遗传因素：精神病中如精神分裂症、情感性精神障碍，家族中精神病的患病率明显高于一般普通人群，而且血缘关系愈近，发病机会愈高。此外，精神发育迟滞、癫痫性精神障碍的遗传性在发病因素中也占相当的比重。这也是精神病的病因之一。 (3)自身：在同样的环境中，承受同样的精神刺激，那些心理素质差、对精神刺激耐受力低的人易发病。通常情况下，性格内向、心胸狭窄、过分自尊的人，不与人交往、孤僻懒散的人受挫折后容易出现精神异常。 (4)躯体因素：感染、中毒、颅脑外伤、肿瘤、内分泌、代谢及营养障碍等均可导致精神障碍，。但应注意，精神障碍伴有的躯体因素，并不完全与精神症状直接相关，有些是由躯体因素直接引起的，有些则是以躯体因素只作为一种诱因而存在。 孕期感染。如果在怀孕期间，孕妇感染了某种病毒，病毒也传染给了胎儿的话，那么，胎儿出生长大后患上精神分裂症的可能性是极其的大。所以怀孕中的女性朋友要注意卫生，尽量不要接触病毒源。 上述就是关于精神分裂症的发病原因，想必大家都已经知道了吧。患上精神分裂症之后，大家也不必过于伤心，现在我国的医疗水平是足以让大家快速恢复过来的，所以说一定要保持良好的情绪。
IX: Introduction to the theory of dynamical systems, stability and bifurcations Parts: 1. Introduction. 2. Diskrete dynamical systems. 3. The Lyapunov exponent. 4. Julia and Mandelbrot sets. 5. Continuous dynamical systems. 6. Some introductory examples. 7. Classification of critical points. 8. The general solution of a linear system. 9. Classification of equilibrium points in specific systems. 10. Exercises.
1. Introduction. A dynamical system is a phenomenon that changes with tim, for instance the position of a pendulum, the weather, the amount of predators and prey in a lake, et cetera. The traditional way of describing a dynamical system is to use a linear system of differential equations. In this case we have a pretty simple theory to solve the problem (see for instace part 8). A more realistic model however often leads to nonlinear systems of differential equations. In this case it is much more complicated to describe the behavios in the long run, but with help of computers and existing theories we can sometimes obtain the solution as an attractor to the system. In many other cases we will instead get bifurcations or chaos. Chaos means that it is hard (or impossible) to determine the long term behavior; small changes in indata gives dramatic changes in the long term behavior. Some attractors can be described as fractals, some particular self similar sets (a small part of the set has the same structure as the whole set). Such attractors are sometimes called strange attractors.
精神分裂症的病因是什么 精神分裂症是一种精神方面的疾病，青壮年发生的概率高，一般 在16～40岁间，没有正常器官的疾病出现，为一种功能性精神病。 精神分裂症大部分的患者是由于在日常的生活和工作当中受到的压力 过大，而患者没有一个良好的疏导的方式所导致。患者在出现该情况 不仅影响本人的正常社会生活，且对家庭和社会也造成很严重的影响。 精神分裂症常见的致病因素： 1、环境因素：工作环境比如经济水平低低收入人群、无职业的人群中，精神分裂症的患病率明显高于经济水平高的职业人群的患病率。还有实际的生活环境生活中的不如意不开心也会诱发该病。 2、心理因素：生活工作中的不开心不满意，导致情绪上的失控，心里长期受到压抑没有办法和没有正确的途径去发泄，如恋爱失败， 婚姻破裂，学习、工作中不愉快都会成为本病的原因。 3、遗传因素：家族中长辈或者亲属中曾经有过这样的病人，后代会出现精神分裂症的机会比正常人要高。 4、精神影响：人的心里与社会要各个方面都有着不可缺少的联系，对社会环境不适应，自己无法融入到社会中去，自己与社会环境不相
适应，精神和心情就会受到一定的影响，大脑控制着人的精神世界， 有可能促发精神分裂症。 5、身体方面：细菌感染、出现中毒情况、大脑外伤、肿瘤、身体的代谢及营养不良等均可能导致使精神分裂症，身体受到外界环境的 影响受到一定程度的伤害，心里受到打击，无法承受伤害造成的痛苦，可能会出现精神的问题。 对于精神分裂症一定要配合治疗，接受全面正确的治疗，最好的 疗法就是中医疗法加心理疗法。早发现并及时治疗并且科学合理的治疗，不要相信迷信，要去正规的医院接受合理的治疗，接受正确的治 疗按照医生的要求对症下药，配合医生和家人，给病人创造一个良好 的治疗环境，对于该病的康复和痊愈会起到意想不到的效果。
Scalable Nonlinear Dynamical Systems for Agent Steering and Crowd Simulation Siome Goldenstein Menelaos Karavelas Dimitris Metaxas Leonidas Guibas Eric Aaron Ambarish Goswami Computer and Information Science Department,University of Pennsylvania. Computer Science Department,Stanford University. Discreet.
1Introduction Modeling autonomous digital agents and simulating their behavior in virtual envi-ronments is becoming increasingly important in computer graphics.In virtual real-ity applications,for example,each agent interacts with other agents and the environ-ment,so complex real-time interactions are necessary to achieve non-trivial behav-ioral scenarios.Modern game applications require smart autonomous agents with varying degrees of intelligence to permit multiple levels of game complexity.Agent behaviors must allow for complex interactions,and they must be adaptive in terms of both time and space(continuous changes in the environment).Finally,the mod-eling approach should scale well with the complexity of the environment geometry, the number and intelligence of the agents,and the various agent-environment inter-actions. There have been several promising approaches towards achieving the above goal. Many of them,however,are restrictive in terms of their application domain.They do not scale well with the complexity of the environment.They do not model time explicitly.They do not guarantee that the desired behavior will always be exhib-ited.This paper presents an alternative:a scalable,adaptive,and mathematically rigorous approach to modeling complex low-level behaviors in real time. We employ nonlinear dynamical system theory,kinetic data structures,and har-monic functions in a novel three-layer approach to modeling autonomous agents in a virtual environment.The?rst layer consists of differential equations based on nonlinear dynamic system theory,modeling the low-level behavior of the au-tonomous agent in complex environments.In the second layer,the motions of the agents,obstacles,and targets are incorporated into a kinetic data structure,provid-ing an ef?cient,scalable approach for adapting an agent’s motion to its changing local environment.In the third layer,differential equations based on harmonic func-tions determine a global course of action for an agent,initializing the differential equations from the?rst layer,guiding the agent,and keeping it from getting stuck in local minima.We also discuss how hybrid systems concepts for global planning can capitalize on both our layered approach and the continuous,reactive nature of our agent steering. In the?rst layer,we characterize in a mathematically precise way the behavior of our agents in complex dynamic virtual environments.The agents exist in a real-time virtual environment consisting of obstacles,targets,and other agents.Depending on the application,agents reach one or multiple targets while avoiding obstacles; targets and obstacles can be stationary and/or moving.Further,the inclusion of time as a variable in our system makes the formulation ef?cient,natural and powerful compared to traditional AI approaches. Our agent modeling is based on the coupling of a set of nonlinear dynamical sys-
卫生部办公厅关于印发《脐带血造血干细胞治疗技术管理规 范(试行)》的通知 【法规类别】采供血机构和血液管理 【发文字号】卫办医政发[2009]189号 【失效依据】国家卫生计生委办公厅关于印发造血干细胞移植技术管理规范(2017年版)等15个“限制临床应用”医疗技术管理规范和质量控制指标的通知 【发布部门】卫生部(已撤销) 【发布日期】2009.11.13 【实施日期】2009.11.13 【时效性】失效 【效力级别】部门规范性文件 卫生部办公厅关于印发《脐带血造血干细胞治疗技术管理规范（试行）》的通知 （卫办医政发〔2009〕189号） 各省、自治区、直辖市卫生厅局，新疆生产建设兵团卫生局： 为贯彻落实《医疗技术临床应用管理办法》，做好脐带血造血干细胞治疗技术审核和临床应用管理，保障医疗质量和医疗安全，我部组织制定了《脐带血造血干细胞治疗技术管理规范（试行）》。现印发给你们，请遵照执行。 二〇〇九年十一月十三日
脐带血造血干细胞 治疗技术管理规范（试行） 为规范脐带血造血干细胞治疗技术的临床应用，保证医疗质量和医疗安全，制定本规范。本规范为技术审核机构对医疗机构申请临床应用脐带血造血干细胞治疗技术进行技术审核的依据，是医疗机构及其医师开展脐带血造血干细胞治疗技术的最低要求。 本治疗技术管理规范适用于脐带血造血干细胞移植技术。 一、医疗机构基本要求 （一）开展脐带血造血干细胞治疗技术的医疗机构应当与其功能、任务相适应，有合法脐带血造血干细胞来源。 （二）三级综合医院、血液病医院或儿童医院，具有卫生行政部门核准登记的血液内科或儿科专业诊疗科目。 1.三级综合医院血液内科开展成人脐带血造血干细胞治疗技术的，还应当具备以下条件： （1）近3年内独立开展脐带血造血干细胞和（或）同种异基因造血干细胞移植15例以上。 （2）有4张床位以上的百级层流病房，配备病人呼叫系统、心电监护仪、电动吸引器、供氧设施。 （3）开展儿童脐带血造血干细胞治疗技术的，还应至少有1名具有副主任医师以上专业技术职务任职资格的儿科医师。 2.三级综合医院儿科开展儿童脐带血造血干细胞治疗技术的，还应当具备以下条件：
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 55, NO. 2, FEBRUARY 2008
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Adaptive Feedback Synchronization of a General Complex Dynamical Network With Delayed Nodes
Qunjiao Zhang, Junan Lu, Jinhu Lü, Senior Member, IEEE, and Chi K. Tse, Fellow, IEEE
Abstract—In the past decade, complex networks have attracted much attention from various ?elds of sciences and engineering. Synchronization is a typical collective behavior of complex networks that has been extensively investigated in recent years. To reveal the dynamical mechanism of synchronization in complex networks with time delays, a general complex dynamical network with delayed nodes is further studied. Based on a suitable model, we investigate the adaptive feedback synchronization and obtain several novel criteria for globally exponentially asymptotic synchronization. In particular, our hypotheses and the proposed adaptive controllers for network synchronization are very simple and can be readily applied in practical applications. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed synchronization criteria. Index Terms—Adaptive feedback synchronization, complex networks, delayed nodes.
I. INTRODUCTION HE so-called complex network refers to a set of nodes connected by edges (graph) that has certain nontrivial topological features that are not found in simple networks [1]–[4]. Such nontrivial features involve a degree distribution with a heavy-tail, a hierarchical structure, a high clustering coef?cient, a community structure at different scales, and assortativity or disassortativity among vertices [2], [5]–[8]. It is well known that complex networks exist in many natural and man-made systems, e.g., food webs, neural networks, cellular and metabolic networks, electrical power grids, computer networks, technological networks, the World Wide Web, coauthorship and citation networks, social networks, etc. [1], [2]. Time delay inevitably exists in natural and man-made networks [9]–[15]. In much of the literature, time delays in the cou-
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Manuscript received May 25, 2007; revised September 17, 2007. This work was supported by National Natural Science Foundation of China under Grants 60574045, 70771084, 60221301 and 60772158, by National Basic Research (973) Program of China under Grant 2007CB310800 and 2007CB310805, by Important Direction Project of Knowledge Innovation Program of Chinese Academy of Sciences under Grant KJCX3-SYW-S01, and by Scienti?c Research Startup Special Foundation on Excellent PhD Thesis, and Presidential Award of Chinese Academy of Sciences. This paper was recommended by Associate Editor J. Suykens. Q. Zhang and J. Lu are with the School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China (e-mail: qunjiao99@https://www.doczj.com/doc/7b18086411.html,, jalu@https://www.doczj.com/doc/7b18086411.html,). J. Lü is with the Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China, and also with the State Key Laboratory of Software Engineering, Wuhan University, Wuhan 430072, China (e-mail: jhlu@https://www.doczj.com/doc/7b18086411.html,). C. K. Tse is with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, China (e-mail: encktse@polyu. edu.hk). Digital Object Identi?er 10.1109/TCSII.2007.911813
plings (edges) are considered [9]–[11]; however, the time delays in the dynamical nodes [12]–[15], which are more complex, are still relatively unexplored. As a matter of fact, one can ?nd numerous examples in the real world which are characterized by delayed differential equations having time delays in the dynamical nodes [12]–[15]. For example, the delayed logistic differential equation, which has time delay in the dynamical node, is a representative dynamical model of the electrochemical intercalations and physiological systems [15]. It is thus imperative to further investigate complex dynamical networks with delayed nodes. However, such complex networks are still relatively unexplored due to their complexity and the absence of an appropriate simpli?cation procedure [9], [10]. Further, the lack of a general approach or tool to study such kind of complex networks has also obstructed the progress of development of their analysis [11]. Recently, we developed a method to deal with such kind of complex networks [16], and in this paper we further investigate the synchronization of a general complex dynamical network with delayed nodes. Synchronization is now widely regarded as a kind of collective behavior which is exhibited in many natural systems [1], [16], [17]. In essence, synchronization is a form of self-organization. It has been demonstrated that many real-world problems have close relationships with network synchronization [1], [2], [8]. For example, theoretical and experimental results show that a mammalian brain not only displays in its storage of associative memories, but also modulates oscillatory neuronal synchronization by selective perceive attention [18]. Recently, synchronization of complex dynamical networks has been a focus in various ?elds of science and engineering. Wu [5] investigated the synchronization of random directed networks. Lü and Chen [8] studied the synchronization of time-varying complex dynamical networks. Li et al. [9], [11] explored the synchronization of complex dynamical networks with nonlinear inner-coupling functions and time delays. Zhou et al. [16] studied the adaptive synchronization of an uncertain complex dynamical network. Sorrentino et al. [17] investigated the controllability of complex networks with pinning controllers. However, the important issue of synchronization of complex dynamical networks with delayed nodes has only been lightly covered [9]–[15]. This paper will further investigate the adaptive feedback synchronization of complex dynamical networks with delayed nodes. In particular, we obtain several novel criteria for globally exponentially asymptotic synchronization. It should be pointed out that our hypotheses and the proposed adaptive controllers for network synchronization are very simple and easy to apply. This paper is organized as follows. Section II introduces a general complex dynamical network with delayed nodes and several useful hypotheses. A set of novel adaptive feedback synchronization criteria are given in Section III. Section IV uses two
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